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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In category theory, a branch of mathematics, the opposite category or dual category C op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself.
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category C op.Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite ...
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness . If an ordered set S {\displaystyle S} has the property that every nonempty subset of S {\displaystyle S} having an upper bound also has a least upper ...
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: : where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map : in the category theory sense.
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite of A to B). [1] If there exists an antiisomorphism between two structures, they are said to be antiisomorphic.
In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids , groups, rings , and algebras can be viewed as categories with a single object.